: Consider a system whose initial state at $t=0$ is given in terms of a complete and orthonormal

set of four vectors $|{}_1$: as follows:

$|(0)$:

where $A$ is a real constant.

$($a$)$ Find $A$ so that $|(0)$: is normalized.

$($b$)$ If the energies corresponding to $|{}_1$: are given by $E_1,E_2,E_3,$ and $E_4,$

respectively, write down the state of the system $|(t)$: at any later time $t.$

Write a python program to: Take ket$-$vectors as inputs and form the ground

state as in the above task. The program should find the energy levels and form

the generalized time$-$dependent state and graph the states at any time $\text{'}t\text{'}.$

Write a short note on time evolution of states in a Finfet.

$V(x)=\{\begin{array}{c}{V}_0\text{;}x>0\\ 0\text{;}x<mn\; id="EditorElement\_2098765">0</mn>\end{array}$

A current of particles travels from $-$ towards right. Write the stationary solutions for each of the

regions. What is the condition that no current returns towards $+.$ Use the matching condition to

express transmission and reflection amplitudes in terms of the incident amplitude.

Write a python program to:Generate a gif to show the reflection and transmission

on either wall as in the above question. The results should be verified with

theoretical results. Write a short note on the matching condition for transmission

and reflection coefficients.